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y=arctan(-6x)

(dy)/(dx)=?
Choose 1 answer:
(A)
(1)/(-6(1+36x^(2)))
(B)
(1)/(-6(1-6x))
(C)
(-6)/(1+36x^(2))
(D)
(-6)/(1-6x)

$y=\arctan (-6 x)$ $\newline$ $\frac{d y}{d x}=?$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{-6\left(1+36 x^{2}\right)}$ $\newline$ (B) $\frac{1}{-6(1-6 x)}$ $\newline$ (C) $\frac{-6}{1+36 x^{2}}$ $\newline$ (D) $\frac{-6}{1-6 x}$

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Find derivatives of inverse trigonometric functions

Full solution

Q.

y=\arctan (-6 x)

\newline

\frac{d y}{d x}=?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{-6\left(1+36 x^{2}\right)}

\newline

(B)

\frac{1}{-6(1-6 x)}

\newline

(C)

\frac{-6}{1+36 x^{2}}

\newline

(D)

\frac{-6}{1-6 x}

Recall Derivative of arctan(x): First, let's recall the derivative of $\text{arctan}(x)$ which is $\frac{1}{1+x^2}$ . Now we need to apply the chain rule because we have $\text{arctan}$ of a function $(-6x)$ , not just $x$ .
Apply Chain Rule: Using the chain rule, the derivative of $\arctan(u)$ with respect to $x$ is $\frac{1}{1+u^2}$ $\cdot$ $\frac{du}{dx}$ , where $u = -6x$ in our case.
Find Derivative of $u$ : Now, let's find the derivative of $u = -6x$ with respect to $x$ , which is simply $\frac{du}{dx} = -6$ .
Substitute into Chain Rule Formula: Substitute $u = -6x$ and $\frac{du}{dx} = -6$ into the chain rule formula to get the derivative of $y$ with respect to $x$ : $\frac{dy}{dx} = \frac{1}{1+(-6x)^2} \times (-6)$ .
Simplify Expression: Simplify the expression: $\frac{dy}{dx} = \frac{-6}{1+36x^2}$ .

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